I just finished my first game of Civilization V -- something for which I even had to buy and build a completely new computer. :P The first thing you notice about the game is that it looks pretty darn nice. The water, the landscape, etc. The first thing I noticed about Civilization IV was that it looked like a round, colorful candy bar. This is not the case with Civ5, which is good. :)
The second thing I noticed about Civ5 was how familiar it felt, even with the hexagonal tiles. You still build cities and move units around just like before. Probably the third thing I noticed was that I had started using the mouse for moving the units. In the previous games I had been playing with the numpad and it could be done in Civ5 too (using numpad 7, 4, 1, 3, 6 and 9), but the mouse feels like more natural way of entering orders in the hex grid.
So, I figured that since I never achieved a cultural victory in Civ4 I'd go for that. I noticed that Egyptians have the special ability of building wonders faster and I figured wonders would produce lots of culture that I needed for cultural victory so I chose to play as Ramesses II. I chose a small map of continents and warlord difficulty level.
I found out there were two city-states to the east from me and two to the west (Oslo and some other). Between the two at the west was a passage to American territory, and behind them was the Siamese civilization. I soon learned that the city-states frequently give you some tasks, such as "destroy that other city-state". Oslo came up with a request like that about its neighbor, and just to try out the new combat mechanics I decided to declare war at the other city.
Now, note that the tasks the city-states give out are not "accept now or forget about it forever" kind of tasks, they just hang out in the air until someone fulfills them. That's quite nice.
So anyway, I attacked the city with some archers, warriors and war chariots. I learned that cities in Civ5 can defend themselves: if I left my units within two tiles of the city, they were sure to sling huge rocks at them. Luckily my warrior got some nice promotion for rough terrain and I fortified it on a hill right next to the city. The city then focused on attacking the warrior which didn't take much damage, while my archers softened up the city until the warrior could just march in.
When you conquer a city state you have three choices: you can annex the city, i.e. it will become your own. This will, however, bring a considerable amount of unhappiness to your empire for quite some time. You can also make the city a puppet, which means that you will get all the benefits from it and it won't be too unhappy, but on the other hand you cannot choose what the city builds nor how it uses its citizens. The final choice is of course to raze the city, which will set the city on fire. Then the city will burn down one citizen a turn until it is completely destroyed. During this time another civ may take it back and put out the flames (this indeed happened to me later).
So anyway, initially I made the city my puppet, but when Washington started behaving threateningly I annexed the city as it was right there at our border. Initially Washington said I shouldn't build my cities near them, but he didn't seem to like me conquering new cities next to them either. They also destroyed the Siamese civilization which had only one city. At about this point I started manning my border with new units. This didn't go unnoticed from Washington who questioned me about it two or three times. Washington also conquered Oslo and then declared war on me.
It turned out Washington hadn't thought it through. I had better technology at him and my musketmen annihilated their swordsmen and pikemen. During the war I also got my musketmen upgraded to riflemen and for the last few cities I even managed to bring in the artillery. The first two cities I took from them I razed as I didn't want to leave my troops to defend them. Once it became clear that Washington was no match to me I started making their cities my puppets. Once I took their capital Washington offered me all but one of their cities for peace but I refused the offer and conquered the rest of their cities as well.
I had pretty much neglected the branch in the technology tree which provided naval units and never settled to other continents. Also, in my main continent I had just six cities which I controlled completely and probably about ten puppet cities. This kept the cost of new social policies low as the required amount of culture increases per each city you own.
At some point I realized it pays to be friends or allies with the city states. Maritime states provide you food and cultured states provide you culture, which I needed to get more social policies. Militaristic states provide you units every once in a while. Also, the states are more likely to vote for you in the United Nations vote (once you get that far) if you are allied with them.
So anyway, I was collecting the social policies in a good pace but my science was advancing even faster and I eventually developed the technologies required for the spaceship. Well, you have to have a plan B so naturally I started building the required parts (some after creating spaceship factories first). I completed the spaceship around 1970s but delayed putting the pieces together for two or three turns to get the final social policy of the fifth branch. Then it was revealed that it would've taken 28 turns to create the Utopia Project so I just launched the spaceship winning the game. :)